Least Common Multiple (LCM) Calculator

1. Enter two or more whole numbers. Type or paste them with commas, spaces or line breaks between them.
2. Check the count. The "Numbers read" line shows how many values the tool found, so a stray comma cannot slip past.
3. Read the LCM and the working. The prime factors of each number, the LCM as prime powers and the pairwise GCD steps appear under the result.

The least common multiple of a set of whole numbers is the smallest positive number that is a multiple of each of them. The NIST Dictionary of Algorithms and Data Structures gives the example LCM(6, 10) = 30: 30 is the first number that both 6 and 10 divide.

A multiple of a number is that number times a whole number, so the multiples of 6 are 6, 12, 18, 24, 30 and so on. A common multiple appears in every list. There are infinitely many common multiples (30, 60, 90 for 6 and 10), and the LCM is the least of them.

The LCM is the mirror image of the greatest common divisor (GCD), also called the highest common factor. The GCD is the biggest number that goes into all the inputs; the LCM is the smallest number that all the inputs go into. You can compute the GCD on its own with the GCD calculator.

Students meet the LCM first as the least common denominator when adding fractions such as 1/12 + 1/18.

The tool uses the prime factorization method, then confirms the answer with the GCD relation.

1. Factor each number into primes by trial division: 12 = 2² x 3, 18 = 2 x 3², 30 = 2 x 3 x 5.
2. List every prime that appears anywhere: 2, 3 and 5.
3. Take the highest exponent of each prime: 2², 3² and 5¹.
4. Multiply them: 4 x 9 x 5 = 180.
5. Check with the GCD. The GCD uses the lowest exponent of the primes common to all numbers, here 2 x 3 = 6. For more than two numbers, the tool applies LCM(a, b) = a x b / GCD(a, b) one pair at a time, and both routes must agree.

University of Maryland number theory notes prove the relation LCM(a, b) x GCD(a, b) = ab for any positive integers a and b. It holds for two numbers at a time, not for three or more at once.

Three buses leave a depot together. One returns every 12 minutes, one every 18 minutes and one every 30 minutes. When do all three next meet?

LCM = 4 x 9 x 5 = 180. The buses meet again after 180 minutes, or 3 hours. By then the first bus has made 15 trips, the second 10 and the third 6.

Pairwise check: LCM(12, 18) = 12 x 18 / 6 = 36, then LCM(36, 30) = 36 x 30 / 6 = 180. Same answer.

A two-number check with 24 and 90: the GCD is 6, so the LCM is 24 x 90 / 6 = 2,160 / 6 = 360. That matches 2³ x 3² x 5 = 360.

Shared prime factors, the size of the numbers and special values such as 1 and 0 shape the result.

Shared Prime Factors

The more primes the numbers share, the smaller the LCM is compared with their product. 12 and 18 share 2 x 3, so their LCM is 36, not 216.

Numbers with No Common Factor

When no two numbers share a prime, the LCM is their product. LCM(3, 5, 7) = 105. Numbers like these are called pairwise coprime.

One Number Divides the Others

When the largest number is a multiple of all the rest, it is the LCM. LCM(4, 8, 16) = 16.

The Number 1 and Duplicates

1 divides everything, so it never changes the LCM. Typing the same number twice also has no effect.

Zero and Negatives

A 0 in the list makes 0 the only common multiple, so the tool returns 0 with a warning. Negative signs are dropped, because the LCM is taken as a positive number.

The LCM and the GCD come from the same prime factorization; one takes the highest powers and the other the lowest.

Three Ways to Find the LCM

Listing multiples works for small numbers but grows slow. Prime factorization shows the structure and handles many numbers at once. The GCD route is fastest for two large numbers, because the Euclid algorithm finds the GCD without factoring.

Adding and Subtracting Fractions

To add 1/12 + 1/18, use the LCM 36 as the common denominator: 3/36 + 2/36 = 5/36. The "LCM divided by each number" line gives the 3 and the 2. For the full fraction arithmetic, a fraction calculator does the rest.

Repeating Schedules

Events on different cycles line up at the LCM of the cycle lengths: buses, shift rotations, maintenance intervals or two blinking lights.

Buying in Packs

Buns come in packs of 8 and hot dogs in packs of 10. The smallest equal count is LCM(8, 10) = 40: 5 packs of buns and 4 packs of hot dogs.

Gears and Rhythms

Two gears with 12 and 18 teeth return to the same starting mesh after 36 teeth pass, which is 3 turns of the small gear and 2 of the large one. Musicians use the same idea for polyrhythms.

1. Multiplying the Numbers Together

12 x 18 = 216 is a common multiple, but not the least. The product is the LCM only when the numbers share no prime factor.

2. Using the Lowest Powers

Taking the lowest exponents gives the GCD, not the LCM. For the LCM, every prime appears at its highest power.

3. Dropping a Prime

A prime that appears in only one number still belongs in the LCM. Leaving out the 5 from 30 would give 36, which 30 does not divide.

4. Applying LCM X GCD = Product to Three Numbers

For 12, 18 and 30 the LCM is 180 and the GCD is 6, but 180 x 6 = 1,080 while 12 x 18 x 30 = 6,480. The relation only holds for two numbers.

5. Stopping the Multiples List Too Soon

When listing multiples by hand, extend both lists until a match appears. For 24 and 90 the first match is 360, the 15th multiple of 24.

6. Factoring Large Numbers Wrongly

A slip in one factorization spoils the answer. Check each factorization by multiplying it back out, or raise primes to powers with the exponent calculator.

Results are exact: the LCM is built with arbitrary-size whole-number arithmetic, so even very large LCMs have no rounding.

Perfect powers show up in the factor list too; a factor such as 2³ x 3³ means the number is a perfect cube, which the cube root calculator confirms.

It is 24. 6 = 2 x 3 and 8 = 2^3, so the LCM is 2^3 x 3 = 24.

It is 2,520, the smallest number that 1, 2, 3 and every whole number up to 10 divide evenly. It equals 2^3 x 3^2 x 5 x 7.

Multiply the two numbers and divide by their GCD. For 24 and 90, 2,160 / 6 = 360.

No. The LCM is a multiple of every input, so it is at least as large as the largest one.

It is their product, because two different primes share no factor. LCM(7, 11) = 77.

Yes, when the numbers are fraction denominators. The least common denominator of 1/4 and 1/6 is LCM(4, 6) = 12.

Factor all three and take each prime at its highest power, or find the LCM of two and then of that result with the third.

By convention it is 0, since 0 is a multiple of every number. Most courses only ask for the LCM of nonzero numbers.

Not directly. Scale the decimals to whole numbers first, for example 0.4 and 0.6 become 4 and 6 with LCM 12, then scale back to 1.2.

No. The calculation runs in your browser, and Save keeps a copy only in your own browser storage.

• Prime factorization and the least common multiple (OpenStax Prealgebra 2e, section 2.5).
• LCM of a set of numbers using the largest exponent of each prime (OpenStax Contemporary Mathematics, section 3.1).
• Proof that LCM(a, b) x GCD(a, b) = ab (University of Maryland, Department of Mathematics).
• Least common multiple definition, LCM(6, 10) = 30 (NIST Dictionary of Algorithms and Data Structures).